The sine addition formula
can look like a formula that must simply be memorized.
But there is a useful geometric technique behind it.
This article focuses on just one part of the derivation:
How does the original horizontal component, cos a, contribute to the final vertical component when the angle is increased by b?
Once this technique is understood, the appearance of the term cos a sin b becomes much more natural.
1. Start with the original angle a
Consider a unit circle with centre O.
Draw a radius making an angle a with the horizontal.
Because the radius has length 1, it can be divided into two components:
- Horizontal component = cos a
- Vertical component = sin a
So the horizontal component has length:
cos a
This is the starting quantity we want to work with.
2. The question we want to answer
Now suppose the original direction is rotated anticlockwise through an additional angle b.
The final direction is therefore:
The final point has a vertical coordinate equal to:
But we are not going to use the addition formula yet.
Instead, we ask:
What contribution does the original horizontal component, cos a, make to this final vertical component?
This is the key technique.
3. Use the component as a new radius
Rather than trying to rotate the original horizontal line directly, we can use its length as the radius of an auxiliary circle.
The original horizontal component has length:
cos a
So construct a new circle with:
radius = cos a
This is the important technique:
Take a component that we already know and use its magnitude as the radius of a new auxiliary circle.
We are not changing the value of the component.
We are simply creating a convenient geometric construction for studying what happens when that component is associated with the additional angle b.
4. Rotate the new radius through b
Take a radius of this smaller circle along the horizontal direction.
Its length is:
cos a
Now rotate that radius anticlockwise through angle b.
We obtain a right triangle.
Here:
and
The vertical side FT is the vertical component produced by this rotated radius.
This is exactly the quantity we are looking for.
5. Identify the sides relative to b
Now use the basic definition of sine.
For angle b:
- Hypotenuse = OF
- Opposite = FT
- Adjacent = OT
Therefore:
So:
But we deliberately constructed:
Therefore:
Rearranging:
So:
6. Why multiplication appears
This technique also explains something that can initially seem strange.
Why don’t we add cos a and sin b?
Because they represent different things.
cos a is a length.
sin b is a ratio.
In this triangle, sin b tells us what fraction of the radius becomes vertical.
So:
vertical component = radius × vertical fraction
The radius is:
cos a
The vertical fraction is:
sin b
Therefore:
7. A numerical example
Suppose, just for illustration, that:
and:
Then a radius of length 10 is being tilted through an angle whose sine is 0.5.
Its vertical component is:
So the vertical contribution is 5.
In the general case, the same calculation is:
Therefore:
8. Why this contribution becomes part of sin(a+b)
We now return to the original unit circle.
The original radius had two components:
Horizontal component = cos a
Vertical component = sin a
After increasing the angle by b, both components contribute to the final vertical position.
We have now calculated the contribution from the original horizontal component:
There will also be a contribution from the original vertical component.
That second contribution will be:
Therefore, the final vertical component will eventually be the sum:
And because the final direction is a+b, that final vertical component is:
Thus:
9. The technique is more important than the formula
The most useful lesson here is not merely:
Instead, it is the method used to obtain it.
The technique
1. Identify a known component.
Here:
horizontal component = cos a
2. Treat its magnitude as the radius of an auxiliary circle.
So:
3. Apply the additional angle b.
Rotate the radius through angle b.
4. Construct the corresponding right triangle.
5. Identify opposite and hypotenuse relative to b.
6. Apply the sine definition.
7. Solve for the required component.
This technique transforms an apparently mysterious algebraic term into a simple geometric calculation.
10. An important distinction
We are not saying that cos a itself becomes sin b.
And we are not saying:
Instead:
cos a tells us the size of the component we are starting with.
sin b tells us the fraction of that component appearing vertically after the direction is tilted through b.
Therefore:
starting component × vertical fraction
The bigger picture
The original unit radius at angle a contains:
Horizontal component → cos a
Vertical component → sin a
When the direction is increased by b, we examine how each component contributes to the final vertical direction.
For the horizontal component:
For the vertical component:
Adding these two vertical contributions eventually gives:
The important conceptual breakthrough is that the two terms are not arbitrary products. Each one comes from applying the basic right-triangle definitions to a component that already exists.
That is the technique that allows the addition formula to be reconstructed rather than memorized.
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